Chi-squared tests for evaluation and comparison of asset pricing models Nikolay Gospodinov, Raymond Kan, and Cesare Robotti
Not-for-publication appendix
In this appendix, we provide additional tests and simulation results that are not included in the paper. We use the same notation as in the paper.
1
Tests for model specification and pairwise comparison
1.1
HJ-distance test of model specification
The following lemma presents the asymptotic distributions of the sample squared HJ-distance under correctly specified and misspecified models. Lemma B.1. Under Assumptions A, B and C in the paper,
(a) if δ = 0, 2 A
T ˆδ ∼ Fn−k (ξ),
(1)
where the ξ i ’s are the eigenvalues of 1
1
A = P 0 U − 2 SU − 2 P,
(2) 1
with P being an n × (n − k) orthonormal matrix whose columns are orthogonal to U − 2 D. (b) if δ > 0, √ where σ2b =
P∞
j=−∞
2 A T (ˆδ − δ 2 ) ∼ N (0, σ2b ),
(3)
E[btbt+j ] and bt = φt (θ ∗ ) − δ 2 .
Proof. See Appendix. The asymptotic distribution and matrix A in part (a) of Lemma B.1 coincide with the ones derived by Jagannathan and Wang (1996) and Parker and Julliard (2005) for the case of linear and nonlinear models, respectively. The asymptotic normality in part (b) of Lemma B.1 has been established by Hansen, Heaton and Luttmer (1995). To conduct inference, the covariance matrices in Lemma B.1 should be replaced with consistent estimators. In particular, in part (a), we can replace A with its sample analog ˆ − 12 SˆU ˆ − 12 Pˆ , Aˆ = Pˆ 0 U 1
(4)
where Sˆ is obtained using a nonparametric heteroskedasticity and autocorrelation consistent (HAC) estimator (see, for example, Newey and West, 1987 and Andrews, 1991), Pˆ is an orthonormal matrix P h ∂yt (ˆγ ) i ˆ with D ˆ = 1 T ˆ − 12 D . Similarly, in part (b) we can whose columns are orthogonal to U t=1 xt ∂γ0 T use a HAC estimator to estimate the variance σ 2b .
1.2
Joint tests of correct specification of two non-nested models
Let F = {ytF (γ F ) ; γ F ∈ ΓF } and G = {ytG (γ G ) ; γ G ∈ ΓG } be two competing models with k1
and k2 parameter vectors and population squared HJ-distances δ 2F = minγ F maxλF E[φF t (θF )] and δ 2G = minγ G maxλG E[φGt (θG )], respectively. To test H0 : δ 2F = δ 2G = 0 for non-nested models, we 2
2
can use the test statistic T (ˆδ F − ˆδ G ) based on the difference of the sample HJ-distances of models
F and G. Alternatively, we can employ an LM test that measures the distance of the Lagrange multipliers of the two models from zero. This will provide a joint test of correct model specification for models F and G. G ∗ G ∗ F ∗ ∗ To set up the notation, define eF t (γ F ) = xt yt (γ F ) − qt−1 , et (γ G ) = xt yt (γ G ) − qt−1 , and
S≡
"
SF
SFG
SGF
SG
#
=
∞ X
j=−∞
E e˜t e˜0t+j ,
(5)
G ∗ 0 0 ∗ 0 where e˜t = [eF t (γ F ) , et (γ G ) ] . Let PF and PG denote orthonormal matrices with dimensions 1
1
n × (n − k1 ) and n × (n − k2 ) whose columns are orthogonal to U − 2 DF and U − 2 DG , respectively,
where DF (DG ) is the D matrix for model F (G ) defined in the paper. Also, denote by PˆF , PˆG ,
ˆ F , and λ ˆ G the sample counterparts of PF , PG , SF , SG , SFG , SGF , λF and λG , SˆF , SˆG , SˆFG , SˆGF , λ respectively. Finally, let ˜ FG = λ
"
ˆ 21 λ ˆF PˆF0 U 1 ˆG ˆ 2λ Pˆ 0 U G
#
.
(6)
The following lemma provides the appropriate asymptotic distribution of the difference in the sample squared HJ-distances when both models are correctly specified and an LM test of H0 : λF = λG = 0n (which is equivalent to testing H0 : δ 2F = δ 2G = 0). Lemma B.2. Suppose that Assumptions A, B and C in the paper hold for each model and ytF (γ ∗F ) 6=
ytG (γ ∗G ). Then, under H0 : δ 2F = δ 2G = 0,
2
(a) 2 A 2 T (ˆδF − ˆδ G ) ∼ F2n−k1 −k2 (ξ),
(7)
where the ξ i ’s are the eigenvalues of the matrix "
1
1
1
PF0 U − 2 SF U − 2 PF 1
1
−PF0 U − 2 SFG U − 2 PG
1
1
PG0 U − 2 SGF U − 2 PF
1
−PG0 U − 2 SG U − 2 PG
#
(8)
,
(b) A 2 ˜0 Σ ˆ −1 λ ˜ FG ∼ LMλ˜ F G = T λ χ2n−k1 −k2 , FG ˜
(9)
λF G
where ˆ˜ = Σ λF G
"
ˆ − 12 SˆF U ˆ − 12 PˆF PˆF0 U ˆ − 21 SˆGF U ˆ − 12 PˆF Pˆ 0 U G
ˆ − 12 SˆFG U ˆ − 12 PˆG PˆF0 U ˆ − 12 SˆG U ˆ − 12 PˆG Pˆ 0 U G
#
.
(10)
Proof. See Appendix. Since the eigenvalues ξ i ’s in part (a) of Lemma B.2 can take on both positive and negative values, the test of the hypothesis H0 : δ 2F = δ 2G = 0 should be two-sided. The LM test in part (b)
of Lemma B.2 offers an alternative way of testing H0 : δ 2F = δ 2G = 0 (using the equivalence between
H0 : δ 2F = δ 2G = 0 and H0 : λF = λG = 0n ) but it is easier to implement and is expected to deliver power gains compared to the test in part (a). The reason is that the test in part (a) may have low 2 2 power in finite samples when ˆδ F ≈ ˆδ G 6= 0 although it is still consistent since under the alternative
ˆδ 2 − ˆδ 2 = Op(T −1/2 ) and |T (ˆδ 2 − ˆδ 2 )| → ∞. F G F G
1.3
Tests for pairwise comparison of nested and non-nested models
In this section, we present the tests of H0 : ytF (γ ∗F ) = ytG (γ ∗G ) and H0 : σ 2d = 0 that can be used for h i P ∂φ (θ ∗ ) ∂ 2 E[φF (θ∗ )] both nested and non-nested models. Let HF = ∂θ t∂θ0 F and MF = limT →∞ Var √1T Tt=1 t∂θ F F
F
with HG and MG defined similarly. Marcellino and Rossi (2008) among others show that under
∗ G ∗ H0 : φF t (θF ) = φt (θ G ),
2 A 2 T (ˆδF − ˆδ G ) ∼ F2n+k1 +k2 (ξ),
where the ξ i ’s are the eigenvalues of the matrix 1 −HF−1 MF 2 HG−1 MGF
−HF−1 MFG HG−1 MG 3
(11)
.
(12)
Several remarks regarding this inference procedure are in order. First, estimating the ξ i ’s from the sample counterpart of the matrix in (12) produces more nonzero estimated ξ i ’s than the theory suggests. In addition, the estimated ξ i ’s do not have the same sign. This is problematic because for nested models, the larger model has a smaller sample HJ-distance by construction. By not imposing 2 2 the constraints that the ξ i ’s should have the same sign, the nonnegative test statistic T (ˆδ F − ˆδ G )
is compared with a distribution that can take on both positive and negative values. Theorems 2 and 3 in the paper propose restricted versions of the test in (11) in addition to easier-to-implement chi-squared tests. Alternatively, we can directly test H0 : σ2d = 0. In this case, A
Tσ ˆ 2d ∼ F2n+k1 +k2 (ξ),
(13)
where σ ˆ 2d is a consistent estimator of σ 2d and the ξ i ’s are four times the squared eigenvalues of the matrix in (12) (see Golden, 2003). It can be shown that for nested models (for F ⊂ G), the test in (13) simplifies to A
Tσ ˆ 2d ∼ Fk2 −k1 (ξ),
(14)
where the ξ i ’s are four times the squared eigenvalues of the matrix ˜ G ΨG 0 )−1 ΨG Σγˆ ΨG 0 , (ΨG∗ H ∗ ∗ ∗ G
(15)
˜ G and Σγˆ are defined in the paper. where ΨG∗ , H G Similarly, for overlapping models (F 6⊂ G, G 6⊂ F and H = F ∩ G with a k3 parameter vector),
the restricted version of the test of H0 : σ 2d = 0 is given by A
Tσ ˆ 2d ∼ Fk1 +k2 −2k3 (ξ), where the ξ i ’s are four times the squared eigenvalues of the matrix " # F 0 −1 ˜ −(ΨF 0(k1 −k3 )×(k2 −k3 ) ∗ HF Ψ∗ ) FG 0 ΨFG ˆ F G Ψ∗ . ∗ Σγ ˜ G ΨG 0 )−1 0(k −k )×(k −k ) (ΨG H 2
3
1
∗
3
(16)
(17)
∗
Finally, for strictly non-nested models (F ∩ G = ∅), the restricted version of the test of H0 :
σ 2d = 0 is
A
Tσ ˆ 2d ∼ F2n−k1 −k2 (ξ), 4
(18)
where the ξ i ’s are four times the squared eigenvalues of the matrix
"
1
1
PF0 U − 2 SF U − 2 PF 1
1
PG0 U − 2 SGF U − 2 PF
1
1
−PF0 U − 2 SFG U − 2 PG 1
1
−PG0 U − 2 SG U − 2 PG
#
(19)
,
defined in the previous section.
2
Simulation inputs
Appendix D in the paper provides the details of the simulation designs. In this section, we include the necessary descriptive statistics about the data used to calibrate the parameters in the simulation study. Since the factor means are set equal to zero (see Appendix D in the paper), we only need to provide the population covariance matrix (V22 ), the population cross-covariance matrix between the returns and the factors (V21 ) and the population covariance matrix of the factors (V11). In all simulations, the number of test assets (n) is 26 (gross risk-free rate plus gross returns on the 25 Fama-French size and book-to-market ranked portfolios). In the linear SDF case, the first 13 columns of V22 (the first row and the first column of the matrix are multiplied by 1000) are given by
2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0.0476 −0.0875 −0.0690 −0.0478 −0.0569 −0.0739 −0.0584 −0.0452 −0.0493 −0.0240 −0.0513 −0.0733 −0.0446 −0.0477 −0.0489 −0.0223 −0.0520 −0.0606 −0.0565 −0.0513 −0.0333 −0.0621 −0.0377 −0.0616 −0.0348 −0.0506
−0.0875 0.0236 0.0189 0.0165 0.0154 0.0161 0.0194 0.0159 0.0134 0.0124 0.0132 0.0166 0.0132 0.0113 0.0106 0.0116 0.0145 0.0114 0.0102 0.0099 0.0113 0.0095 0.0084 0.0068 0.0070 0.0079
−0.0690 0.0189 0.0174 0.0148 0.0141 0.0148 0.0165 0.0143 0.0124 0.0117 0.0125 0.0143 0.0120 0.0107 0.0103 0.0112 0.0123 0.0105 0.0095 0.0094 0.0109 0.0083 0.0075 0.0063 0.0069 0.0077
−0.0478 0.0165 0.0148 0.0137 0.0128 0.0136 0.0143 0.0127 0.0112 0.0109 0.0115 0.0123 0.0108 0.0098 0.0094 0.0102 0.0105 0.0095 0.0087 0.0086 0.0099 0.0071 0.0067 0.0056 0.0063 0.0069
−0.0569 0.0154 0.0141 0.0128 0.0125 0.0132 0.0134 0.0121 0.0108 0.0106 0.0113 0.0116 0.0103 0.0095 0.0093 0.0101 0.0099 0.0091 0.0085 0.0085 0.0096 0.0067 0.0064 0.0055 0.0063 0.0070
−0.0739 0.0161 0.0148 0.0136 0.0132 0.0147 0.0138 0.0126 0.0114 0.0113 0.0124 0.0118 0.0107 0.0101 0.0100 0.0111 0.0099 0.0094 0.0090 0.0090 0.0105 0.0067 0.0066 0.0057 0.0068 0.0076
−0.0584 0.0194 0.0165 0.0143 0.0134 0.0138 0.0184 0.0146 0.0124 0.0115 0.0119 0.0158 0.0125 0.0108 0.0101 0.0106 0.0139 0.0113 0.0099 0.0094 0.0106 0.0095 0.0085 0.0067 0.0068 0.0075
5
−0.0452 0.0159 0.0143 0.0127 0.0121 0.0126 0.0146 0.0131 0.0111 0.0106 0.0111 0.0128 0.0109 0.0099 0.0094 0.0099 0.0111 0.0099 0.0089 0.0086 0.0099 0.0077 0.0073 0.0060 0.0065 0.0070
−0.0493 0.0134 0.0124 0.0112 0.0108 0.0114 0.0124 0.0111 0.0104 0.0097 0.0101 0.0109 0.0097 0.0090 0.0088 0.0092 0.0094 0.0087 0.0082 0.0080 0.0091 0.0067 0.0065 0.0054 0.0061 0.0066
−0.0240 0.0124 0.0117 0.0109 0.0106 0.0113 0.0115 0.0106 0.0097 0.0101 0.0102 0.0099 0.0093 0.0089 0.0088 0.0093 0.0085 0.0084 0.0080 0.0081 0.0091 0.0062 0.0062 0.0054 0.0063 0.0067
−0.0513 0.0132 0.0125 0.0115 0.0113 0.0124 0.0119 0.0111 0.0101 0.0102 0.0116 0.0103 0.0096 0.0092 0.0092 0.0102 0.0088 0.0087 0.0084 0.0085 0.0098 0.0063 0.0064 0.0054 0.0065 0.0074
−0.0733 0.0166 0.0143 0.0123 0.0116 0.0118 0.0158 0.0128 0.0109 0.0099 0.0103 0.0147 0.0111 0.0095 0.0088 0.0092 0.0127 0.0100 0.0087 0.0082 0.0092 0.0091 0.0078 0.0062 0.0063 0.0067
−0.0446 0.0132 0.0120 0.0108 0.0103 0.0107 0.0125 0.0109 0.0097 0.0093 0.0096 0.0111 0.0101 0.0088 0.0085 0.0089 0.0099 0.0090 0.0082 0.0079 0.0087 0.0070 0.0068 0.0057 0.0060 0.0065
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
and the last 13 columns are given by 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
−0.0477 0.0113 0.0107 0.0098 0.0095 0.0101 0.0108 0.0099 0.0090 0.0089 0.0092 0.0095 0.0088 0.0087 0.0081 0.0085 0.0084 0.0082 0.0077 0.0075 0.0084 0.0061 0.0061 0.0053 0.0059 0.0063
−0.0489 0.0106 0.0103 0.0094 0.0093 0.0100 0.0101 0.0094 0.0088 0.0088 0.0092 0.0088 0.0085 0.0081 0.0086 0.0087 0.0078 0.0077 0.0077 0.0077 0.0086 0.0057 0.0059 0.0052 0.0062 0.0065
−0.0223 0.0116 0.0112 0.0102 0.0101 0.0111 0.0106 0.0099 0.0092 0.0093 0.0102 0.0092 0.0089 0.0085 0.0087 0.0102 0.0079 0.0080 0.0078 0.0080 0.0094 0.0057 0.0059 0.0052 0.0063 0.0071
−0.0520 0.0145 0.0123 0.0105 0.0099 0.0099 0.0139 0.0111 0.0094 0.0085 0.0088 0.0127 0.0099 0.0084 0.0078 0.0079 0.0121 0.0092 0.0080 0.0075 0.0080 0.0086 0.0074 0.0059 0.0057 0.0061
−0.0606 0.0114 0.0105 0.0095 0.0091 0.0094 0.0113 0.0099 0.0087 0.0084 0.0087 0.0100 0.0090 0.0082 0.0077 0.0080 0.0092 0.0089 0.0078 0.0073 0.0080 0.0067 0.0066 0.0056 0.0059 0.0062
−0.0565 0.0102 0.0095 0.0087 0.0085 0.0090 0.0099 0.0089 0.0082 0.0080 0.0084 0.0087 0.0082 0.0077 0.0077 0.0078 0.0080 0.0078 0.0077 0.0071 0.0078 0.0060 0.0061 0.0052 0.0059 0.0061
−0.0513 0.0099 0.0094 0.0086 0.0085 0.0090 0.0094 0.0086 0.0080 0.0081 0.0085 0.0082 0.0079 0.0075 0.0077 0.0080 0.0075 0.0073 0.0071 0.0076 0.0080 0.0057 0.0057 0.0050 0.0059 0.0062
−0.0333 0.0113 0.0109 0.0099 0.0096 0.0105 0.0106 0.0099 0.0091 0.0091 0.0098 0.0092 0.0087 0.0084 0.0086 0.0094 0.0080 0.0080 0.0078 0.0080 0.0102 0.0060 0.0060 0.0053 0.0065 0.0073
−0.0621 0.0095 0.0083 0.0071 0.0067 0.0067 0.0095 0.0077 0.0067 0.0062 0.0063 0.0091 0.0070 0.0061 0.0057 0.0057 0.0086 0.0067 0.0060 0.0057 0.0060 0.0078 0.0062 0.0050 0.0050 0.0052
−0.0377 0.0084 0.0075 0.0067 0.0064 0.0066 0.0085 0.0073 0.0065 0.0062 0.0064 0.0078 0.0068 0.0061 0.0059 0.0059 0.0074 0.0066 0.0061 0.0057 0.0060 0.0062 0.0063 0.0050 0.0051 0.0053
−0.0616 0.0068 0.0063 0.0056 0.0055 0.0057 0.0067 0.0060 0.0054 0.0054 0.0054 0.0062 0.0057 0.0053 0.0052 0.0052 0.0059 0.0056 0.0052 0.0050 0.0053 0.0050 0.0050 0.0051 0.0046 0.0046
−0.0348 0.0070 0.0069 0.0063 0.0063 0.0068 0.0068 0.0065 0.0061 0.0063 0.0065 0.0063 0.0060 0.0059 0.0062 0.0063 0.0057 0.0059 0.0059 0.0059 0.0065 0.0050 0.0051 0.0046 0.0058 0.0056
−0.0506 0.0079 0.0077 0.0069 0.0070 0.0076 0.0075 0.0070 0.0066 0.0067 0.0074 0.0067 0.0065 0.0063 0.0065 0.0071 0.0061 0.0062 0.0061 0.0062 0.0073 0.0052 0.0053 0.0046 0.0056 0.0073
In addition, V21 (the matrix is multiplied by 1000) is given by −0.1039 6 10.0010 6 6 8.8866 6 6 7.7876 6 6 7.4489 6 7.7372 6 6 9.6234 6 6 8.1399 6 6 7.2145 6 6 6.8046 6 6 7.1070 6 6 8.9141 6 6 7.3653 6 6 6.6115 6 6 6.3383 6 6 6.4669 6 6 8.2503 6 6 7.0069 6 6 6.3969 6 6 6.1369 6 6 6.6610 6 6 6.7358 6 6 6.0179 6 5.1085 6 4 5.2417 5.5297 2
−0.0028 6.6710 5.8508 5.2760 4.9435 5.3229 5.4680 4.7476 4.0510 3.8339 4.1678 4.4117 3.6984 3.2744 3.0177 3.5102 3.3307 2.8412 2.5004 2.5123 3.0569 1.3186 1.2755 0.8960 1.1248 1.4114
0.0190 −3.1832 −1.8130 −1.0787 −0.6097 0.0261 −3.3460 −1.6474 −0.8552 −0.0197 0.3161 −3.3189 −1.3671 −0.4112 0.2536 0.6817 −3.1970 −1.2485 −0.4126 0.0642 0.4514 −2.4705 −1.3505 −0.6450 0.2833 0.5855
−0.0015 −0.0541 −0.0695 −0.0602 −0.0492 −0.0513 −0.0872 −0.0724 −0.0437 −0.0602 −0.0670 −0.0692 −0.0678 −0.0494 −0.0511 −0.0504 −0.0576 −0.0650 −0.0611 −0.0352 −0.0436 −0.0389 −0.0519 −0.0443 −0.0295 −0.0448
−0.0074 0.1961 0.1848 0.1630 0.1536 0.1797 0.1711 0.1458 0.1481 0.1475 0.1657 0.1556 0.1452 0.1384 0.1369 0.1603 0.1341 0.1203 0.1259 0.1423 0.1400 0.1312 0.0962 0.0771 0.1110 0.1109
−0.0060 0.2304 0.2532 0.2339 0.2160 0.2385 0.3355 0.2524 0.2498 0.2785 0.2150 0.3312 0.2759 0.2407 0.2543 0.2153 0.3014 0.2861 0.2336 0.2100 0.2542 0.2781 0.2654 0.2449 0.2390 0.2588
−0.0000 0.0009 0.0011 0.0012 0.0011 0.0015 0.0013 0.0013 0.0013 0.0015 0.0016 0.0012 0.0013 0.0013 0.0015 0.0016 0.0011 0.0015 0.0017 0.0011 0.0018 0.0010 0.0014 0.0013 0.0015 0.0017
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7, 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
where the seven columns correspond, in the order, to rmkt , rsmb , rhml , ∆cdur , ∆cndur , cay, and ∆cndur ∗ cay. For example, for the linear YOGO model, the V21 matrix (multiplied by 1000) would be (26 × 3) (formed by taking columns one, four and five of the matrix above). Finally, the V11 matrix (multiplied by 1000) is given by 6.5794 6 1.8969 6 6 −1.5929 6 6 −0.0443 6 0.1271 6 4 0.2729 0.0012 2
1.8969 3.0605 −0.4002 −0.0147 0.0448 0.0059 −0.0000
−1.5929 −0.4002 2.9897 0.0102 0.0017 −0.0509 0.0004
−0.0443 −0.0147 0.0102 0.0289 0.0065 −0.0322 −0.0001
0.1271 0.0448 0.0017 0.0065 0.0276 −0.0008 −0.0000
0.2729 0.0059 −0.0509 −0.0322 −0.0008 0.1824 0.0007
0.0012 −0.0000 0.0004 −0.0001 −0.0000 0.0007 0.0000
3
7 7 7 7 7, 7 7 5
with the factors in the same order as above. For example, for the linear YOGO model, the V11 matrix (multiplied by 1000) would be (3 × 3), i.e., 6
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7. 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
2
6.5794 4 −0.0443 0.1271
−0.0443 0.0289 0.0065
3 0.1271 0.0065 5 . 0.0276
In the log-linear SDF case, the first 13 columns of V22 (the first row and the first column of the matrix are multiplied by 1000) are given by
2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0.0461 −0.1027 −0.0753 −0.0553 −0.0613 −0.0769 −0.0733 −0.0544 −0.0561 −0.0281 −0.0533 −0.0861 −0.0525 −0.0527 −0.0502 −0.0246 −0.0608 −0.0680 −0.0598 −0.0531 −0.0332 −0.0655 −0.0408 −0.0627 −0.0336 −0.0458
−0.1027 0.0235 0.0185 0.0160 0.0149 0.0155 0.0194 0.0156 0.0133 0.0121 0.0128 0.0168 0.0131 0.0114 0.0104 0.0112 0.0144 0.0116 0.0100 0.0097 0.0109 0.0097 0.0085 0.0070 0.0069 0.0080
−0.0753 0.0185 0.0167 0.0140 0.0134 0.0139 0.0163 0.0138 0.0121 0.0112 0.0119 0.0142 0.0117 0.0105 0.0099 0.0107 0.0121 0.0104 0.0093 0.0091 0.0103 0.0083 0.0075 0.0063 0.0067 0.0076
−0.0553 0.0160 0.0140 0.0129 0.0120 0.0126 0.0140 0.0122 0.0108 0.0103 0.0108 0.0121 0.0104 0.0096 0.0090 0.0097 0.0103 0.0093 0.0084 0.0082 0.0093 0.0071 0.0066 0.0056 0.0061 0.0067
−0.0613 0.0149 0.0134 0.0120 0.0117 0.0122 0.0131 0.0116 0.0103 0.0100 0.0106 0.0114 0.0099 0.0093 0.0088 0.0095 0.0097 0.0089 0.0082 0.0081 0.0090 0.0067 0.0063 0.0054 0.0060 0.0068
−0.0769 0.0155 0.0139 0.0126 0.0122 0.0135 0.0134 0.0120 0.0108 0.0106 0.0115 0.0115 0.0102 0.0098 0.0094 0.0103 0.0097 0.0092 0.0086 0.0085 0.0098 0.0067 0.0065 0.0057 0.0064 0.0073
−0.0733 0.0194 0.0163 0.0140 0.0131 0.0134 0.0185 0.0144 0.0124 0.0112 0.0117 0.0159 0.0125 0.0110 0.0099 0.0104 0.0139 0.0114 0.0098 0.0093 0.0103 0.0097 0.0086 0.0068 0.0068 0.0076
−0.0544 0.0156 0.0138 0.0122 0.0116 0.0120 0.0144 0.0127 0.0109 0.0102 0.0106 0.0128 0.0106 0.0098 0.0091 0.0095 0.0110 0.0099 0.0087 0.0083 0.0094 0.0077 0.0073 0.0060 0.0064 0.0069
−0.0561 0.0133 0.0121 0.0108 0.0103 0.0108 0.0124 0.0109 0.0102 0.0093 0.0097 0.0109 0.0096 0.0090 0.0085 0.0089 0.0094 0.0088 0.0081 0.0078 0.0087 0.0068 0.0065 0.0054 0.0060 0.0065
−0.0281 0.0121 0.0112 0.0103 0.0100 0.0106 0.0112 0.0102 0.0093 0.0095 0.0097 0.0098 0.0090 0.0087 0.0084 0.0088 0.0084 0.0083 0.0077 0.0077 0.0086 0.0061 0.0061 0.0053 0.0060 0.0065
−0.0533 0.0128 0.0119 0.0108 0.0106 0.0115 0.0117 0.0106 0.0097 0.0097 0.0108 0.0102 0.0093 0.0090 0.0088 0.0096 0.0087 0.0085 0.0081 0.0081 0.0092 0.0063 0.0063 0.0054 0.0063 0.0071
−0.0861 0.0168 0.0142 0.0121 0.0114 0.0115 0.0159 0.0128 0.0109 0.0098 0.0102 0.0149 0.0111 0.0096 0.0087 0.0090 0.0127 0.0102 0.0086 0.0082 0.0090 0.0092 0.0078 0.0063 0.0062 0.0068
−0.0525 0.0131 0.0117 0.0104 0.0099 0.0102 0.0125 0.0106 0.0096 0.0090 0.0093 0.0111 0.0099 0.0088 0.0082 0.0085 0.0098 0.0089 0.0080 0.0077 0.0083 0.0070 0.0067 0.0056 0.0059 0.0064
3
−0.0598 0.0100 0.0093 0.0084 0.0082 0.0086 0.0098 0.0087 0.0081 0.0077 0.0081 0.0086 0.0080 0.0076 0.0074 0.0075 0.0079 0.0077 0.0075 0.0069 0.0075 0.0060 0.0060 0.0051 0.0057 0.0060
−0.0531 0.0097 0.0091 0.0082 0.0081 0.0085 0.0093 0.0083 0.0078 0.0077 0.0081 0.0082 0.0077 0.0074 0.0074 0.0077 0.0074 0.0072 0.0069 0.0073 0.0076 0.0056 0.0056 0.0049 0.0056 0.0060
−0.0332 0.0109 0.0103 0.0093 0.0090 0.0098 0.0103 0.0094 0.0087 0.0086 0.0092 0.0090 0.0083 0.0081 0.0081 0.0088 0.0078 0.0077 0.0075 0.0076 0.0095 0.0060 0.0059 0.0052 0.0061 0.0070
−0.0655 0.0097 0.0083 0.0071 0.0067 0.0067 0.0097 0.0077 0.0068 0.0061 0.0063 0.0092 0.0070 0.0062 0.0057 0.0057 0.0086 0.0069 0.0060 0.0056 0.0060 0.0079 0.0062 0.0050 0.0049 0.0052
−0.0408 0.0085 0.0075 0.0066 0.0063 0.0065 0.0086 0.0073 0.0065 0.0061 0.0063 0.0078 0.0067 0.0061 0.0058 0.0058 0.0074 0.0067 0.0060 0.0056 0.0059 0.0062 0.0062 0.0049 0.0050 0.0053
−0.0627 0.0070 0.0063 0.0056 0.0054 0.0057 0.0068 0.0060 0.0054 0.0053 0.0054 0.0063 0.0056 0.0053 0.0051 0.0051 0.0059 0.0057 0.0051 0.0049 0.0052 0.0050 0.0049 0.0050 0.0045 0.0045
−0.0336 0.0069 0.0067 0.0061 0.0060 0.0064 0.0068 0.0064 0.0060 0.0060 0.0063 0.0062 0.0059 0.0058 0.0059 0.0060 0.0057 0.0058 0.0057 0.0056 0.0061 0.0049 0.0050 0.0045 0.0056 0.0054
−0.0458 0.0080 0.0076 0.0067 0.0068 0.0073 0.0076 0.0069 0.0065 0.0065 0.0071 0.0068 0.0064 0.0063 0.0063 0.0068 0.0062 0.0062 0.0060 0.0060 0.0070 0.0052 0.0053 0.0045 0.0054 0.0071
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
and the last 13 columns are given by 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
−0.0527 0.0114 0.0105 0.0096 0.0093 0.0098 0.0110 0.0098 0.0090 0.0087 0.0090 0.0096 0.0088 0.0086 0.0079 0.0083 0.0085 0.0082 0.0076 0.0074 0.0081 0.0062 0.0061 0.0053 0.0058 0.0063
−0.0502 0.0104 0.0099 0.0090 0.0088 0.0094 0.0099 0.0091 0.0085 0.0084 0.0088 0.0087 0.0082 0.0079 0.0082 0.0082 0.0077 0.0076 0.0074 0.0074 0.0081 0.0057 0.0058 0.0051 0.0059 0.0063
−0.0246 0.0112 0.0107 0.0097 0.0095 0.0103 0.0104 0.0095 0.0089 0.0088 0.0096 0.0090 0.0085 0.0083 0.0082 0.0095 0.0078 0.0079 0.0075 0.0077 0.0088 0.0057 0.0058 0.0051 0.0060 0.0068
−0.0608 0.0144 0.0121 0.0103 0.0097 0.0097 0.0139 0.0110 0.0094 0.0084 0.0087 0.0127 0.0098 0.0085 0.0077 0.0078 0.0119 0.0093 0.0079 0.0074 0.0078 0.0086 0.0074 0.0059 0.0057 0.0062
−0.0680 0.0116 0.0104 0.0093 0.0089 0.0092 0.0114 0.0099 0.0088 0.0083 0.0085 0.0102 0.0089 0.0082 0.0076 0.0079 0.0093 0.0089 0.0077 0.0072 0.0077 0.0069 0.0067 0.0057 0.0058 0.0062
7
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7. 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
In addition, V21 (the matrix is multiplied by 1000) is given by −0.0591 6 10.0119 6 6 8.7143 6 6 7.5990 6 6 7.2600 6 7.4879 6 6 9.6422 6 6 8.0169 6 6 7.1312 6 6 6.6456 6 6 6.9387 6 6 8.9031 6 6 7.2469 6 6 6.5775 6 6 6.1914 6 6 6.3149 6 6 8.1799 6 6 6.9831 6 6 6.2619 6 6 5.9680 6 6 6.4578 6 6 6.7234 6 6 5.9461 6 5.0138 6 4 5.1097 5.4627 2
−0.1067 10.2757 8.9235 7.7728 7.4329 7.6788 9.8683 8.1964 7.2987 6.7748 7.0958 9.1308 7.4129 6.7319 6.3347 6.4329 8.3684 7.1618 6.4176 6.1114 6.5863 6.8930 6.0785 5.1537 5.2180 5.5857
−0.0040 6.4540 5.5649 4.9715 4.6344 4.9401 5.3268 4.5564 3.8893 3.6301 3.9055 4.3296 3.5752 3.1840 2.8736 3.3084 3.2484 2.7972 2.4058 2.4021 2.8807 1.3037 1.2648 0.9070 1.0743 1.3680
0.0158 −3.2497 −1.8180 −1.1230 −0.6840 −0.1254 −3.3531 −1.6700 −0.9080 −0.0887 0.1875 −3.3137 −1.3770 −0.4726 0.1554 0.5238 −3.1243 −1.2805 −0.4697 −0.0097 0.3242 −2.4337 −1.3354 −0.6570 0.2292 0.4989
−0.0073 0.2008 0.1843 0.1605 0.1498 0.1747 0.1762 0.1465 0.1477 0.1455 0.1626 0.1629 0.1449 0.1382 0.1346 0.1556 0.1383 0.1217 0.1241 0.1407 0.1369 0.1327 0.0965 0.0750 0.1088 0.1107
−0.0063 0.1252 0.1501 0.1111 0.1170 0.1304 0.0914 0.0932 0.0978 0.0952 0.1157 0.0866 0.0706 0.0822 0.0768 0.0958 0.0603 0.0692 0.0588 0.0679 0.1103 0.0598 0.0442 0.0510 0.0617 0.1024
−0.0014 −0.0560 −0.0687 −0.0559 −0.0471 −0.0461 −0.0843 −0.0672 −0.0416 −0.0547 −0.0604 −0.0670 −0.0652 −0.0456 −0.0467 −0.0466 −0.0568 −0.0625 −0.0566 −0.0317 −0.0391 −0.0344 −0.0501 −0.0430 −0.0275 −0.0425
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7, 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
where the seven columns correspond, in the order, to ln(Rmkt), ln(1+rmkt), ln(1+rsmb), ln(1+rhml ), ∆cndur at time t and t − 1, respectively, and ∆cdur . Finally, the V11 matrix (multiplied by 1000) is given by 6.4069 6 6.5649 6 6 1.8502 6 6 −1.5634 6 6 0.1199 4 0.0616 −0.0428 2
6.5649 6.7740 1.8851 −1.6090 0.1293 0.0690 −0.0414
1.8502 1.8851 2.9867 −0.4156 0.0447 0.0397 −0.0158
−1.5634 −1.6090 −0.4156 2.9453 0.0026 0.0240 0.0094
0.1199 0.1293 0.0447 0.0026 0.0276 0.0095 0.0065
0.0616 0.0690 0.0397 0.0240 0.0095 0.0283 0.0096
−0.0428 −0.0414 −0.0158 0.0094 0.0065 0.0096 0.0289
3
7 7 7 7 7, 7 7 5
with the factors in the same order as above.
3
Additional simulation results
3.1 3.1.1
Multivariate normally distributed factors and returns Nested models
In Table 5, we evaluate the size and power properties of the tests of H0 : ytF (γ ∗F ) = ytG (γ ∗G ) and
H0 : σ 2d = 0 in pairwise nested model comparison tests. The results in this section complement the pairwise nested model comparison results reported in Panels A and C of Table 3 in the paper. Specifically, we consider three additional tests: 1) the unrestricted weighted chi-squared test (UFT) in (11); 2) the unrestricted variance test (UVT) in (13); and 3) the restricted variance test (RVT) in (14). The data and simulation designs are the same as the ones for Panels A and C of Table 3 in the paper. 8
Table 5 about here
Several interesting insights emerge from Table 5: • UFT performs well in terms of size across linear and nonlinear asset pricing specifications. However, it is dominated in terms of power by the chi-squared test in part (b) of Theorem 2 in the paper; • UVT is very conservative under the null and has substantially lower power than all the other tests; • RVT exhibits better size and power properties than UVT but is still dominated in terms of size and power by UFT and especially by the chi-squared test in part (b) of Theorem 2 in the paper. 3.1.2
Non-nested models
In Table 6, we evaluate the size and power properties of the tests of H0 : ytF (γ ∗F ) = ytG (γ ∗G ) and
H0 : σ2d = 0 in pairwise non-nested model comparison tests. The results in this section complement the pairwise non-nested model comparison results reported in Panels A and C of Table 4 in the paper. Specifically, we consider three additional tests: 1) the unrestricted weighted chi-squared test (UFT) in (11); 2) the unrestricted variance test (UVT) in (13); and 3) the restricted variance test (RVT) in (16). The data and simulation designs are the same as the ones for Panels A and C of Table 4 in the paper.
Table 6 about here
Several interesting insights emerge from Table 6: • UFT performs well in terms of size (for T > 120) across linear and nonlinear asset pricing specifications. However, it is strongly dominated in terms of power by the chi-squared test in part (b) of Theorem 3 in the paper. The power of UFT is much lower for nonlinear models than for linear models; 9
• UVT is very conservative under the null and has substantially lower power than all the other tests; • RVT exhibits better size and power properties than UVT but is still dominated in terms of size and power by the chi-squared test in part (b) of Theorem 3 in the paper. In light of the discussion in Section 3 in the paper, if we reject H0 : ytF (γ ∗F ) = ytG (γ ∗G ), we should then test whether models F and G are correctly specified before applying the normal test. In Table 7, we evaluate the size and power properties of the tests of H0 : δ2F = δ 2G = 0 and
H0 : σ 2d = 0 for two overlapping distinct SDFs. We consider five tests: 1) the restricted weighted chi-squared test (RFT) in part (a) of Lemma B.2; 2) the LM test (LM) in part (b) of Lemma B.2; 3) the unrestricted weighted chi-squared test (UFT) in (11); 4) the unrestricted variance test (UVT) in (13); and 5) the restricted variance test (RVT) in (16). To examine size in the linear SDF case, we consider two correctly specified one-factor models. The factor in each model is created by adding a normally distributed error to rmkt. The error
term in each model has a mean of zero and a variance of 20% of the variance of rmkt . The two error terms are independent of each other as well as of the returns and the factor. The implied population HJ-distances of the two models are both equal to zero. To analyze power, we simply set the return means equal to the means estimated from the sample and consider the two models above. This implies that the population HJ-distances of these two models are both equal to 0.6524. To examine size in the log-linear SDF case, we also consider two correctly specified one-factor models. The factor in each model is created by adding a normally distributed error to ln(Rmkt ). The error term in each model has a mean of zero and a variance of 20% of the variance of ln(Rmkt ). The two error terms are independent of each other as well as of the returns and the factor. The implied population HJ-distances of the two models are both equal to zero. To analyze power, we simply set the return means equal to the means estimated from the sample and consider the two models above. This implies that the population HJ-distances of these two models are both equal to 0.6377. Panels A and B of Table 7 are for linear and nonlinear SDFs, respectively. Table 7 about here Several interesting insights emerge from Table 7: 10
• RFT in part (a) of Lemma (B.2) enjoys the best size properties overall, followed by UFT; • LM in part (b) of Lemma (B.2) is severely undersized but has the best power properties among the five tests. Given our simulation design, the underrejection problem under the null for the LM test occurs because the eigenvalues of Σλ˜ F G are of very different magnitudes ˆ ˜ then leads to and, in particular, half of them are very close to zero. Sampling error in Σ λF G ˆ −1 has many abnormally large eigenvalues. Since these large many tiny eigenvalues, and Σ ˜ λF G
˜FG with abnormally low variance, this eigenvalues are associated with linear combinations of λ will lead to a severe underrejection problem for LM in our simulation experiment. • UVT and RVT are conservative under the null but perform well in terms of power for T > 120.
3.2
Multivariate t(8) distributed factors and returns
In Tables 8 and 9, we use the same data and simulation designs used in Tables 1 through 4 in the paper. The only difference is that we assume that the factors and the returns are multivariate t-distributed. The number of degrees of freedom of the t-distribution is set equal to 8 in all simulations. Tables 8 and 9 about here The t-distribution results are very similar to the ones obtained under normality. Tables 8 and 9 show that the only noteworthy difference is a slight increase in the empirical size of the tests in the t-distribution case.
11
Appendix: Proofs Proof of Lemma B.1. (a) From the definition of H in the paper, we can use the partitioned matrix inverse formula to obtain " #−1 " # 0 ˜ ˜ 0 U −1 H HD 2C 2D 1 , H −1 = = ˜ −U −1 + U −1 DHD ˜ 0 U −1 2 U −1 DH 2D −2U
(A.1)
˜ = (C + D 0 U −1 D)−1 . Under the null hypothesis H0 : δ = 0, using Lemma A.1 in the paper where H we have ˆδ 2 = − 1 v¯T (θ∗ )0 H −1 v¯T (θ∗ ) + op 2
1 T
(A.2)
since λ∗ = 0n and φt (γ ∗ , 0n) = E[φt (γ ∗ , 0n)] = 0. Let v¯T (θ∗ ) = [¯ v1,T (θ∗ )0 , v¯2,T (θ ∗ )0 ]0 , where v¯1,T (θ∗ ) denotes the first k elements of v¯T (θ∗ ). Under the null, v¯1,T (θ∗ ) = 0k and C = 0k×k . Then, it follows that 2
T ˆδ
√ 1√ = − T v¯T (θ∗ )0 H −1 T v¯T (θ ∗ ) + op(1) 2 √ 1√ T v¯2T (θ ∗ )0 [U −1 − U −1 D(D 0 U −1 D)−1 D0 U −1 ] T v¯2,T (θ ∗ ) + op(1) = 4 1√ 1 1√ = T v¯2T (θ ∗ )0 U − 2 P P 0 U − 2 T v¯2,T (θ ∗ ) + op (1) 4 1
(A.3)
1
by using the fact that In − U − 2 D(D 0 U −1 D)−1 D0 U − 2 = P P 0 . Also, Assumptions A, B and C in √ the paper ensure that the empirical process T v¯2,T (θ∗ ) obeys the central limit theorem and √ A T v¯2,T (θ ∗ ) ∼ N (0n, Mλλ ).
(A.4)
Thus, using the fact that Mλλ = 4S under the null, we obtain 2 A
1
1
1
1
T ˆδ ∼ z 0 S 2 U − 2 P P 0 U − 2 S 2 z, 1
1
1
1
(A.5) 1
1
where z ∼ N (0n , In). Since S 2 U − 2 P P 0 U − 2 S 2 has the same nonzero eigenvalues as P 0 U − 2 SU − 2 P , we have 2 A
T ˆδ ∼ Fn−k (ξ), 1
(A.6)
1
where the ξ i ’s are the eigenvalues of P 0 U − 2 SU − 2 P . This completes the proof of part (a). √ 2 (b) Now consider the case δ > 0. In this situation, the asymptotic behavior of T (ˆδ − δ 2 ) is P determined by √1T Tt=1 (φt(θ ∗ ) − E[φt (θ∗ )]), which converges weakly to a Gaussian process. Under 12
Assumptions A, B and C in the paper, and since E[φt(θ ∗ )] = δ 2 , we have √
T 2 1 X A 2 ˆ T (δ − δ ) = √ (φt (θ∗ ) − E[φt (θ∗ )]) + op (1) ∼ N (0, σ2b ). T t=1
(A.7)
This completes the proof of part (b). Proof of Lemma B.2. (a) From Lemma A.1 in the paper and under the null H0 : δ 2F = δ 2G = 0, we obtain 2 2 T (ˆδ F − ˆδ G ) " √ F ∗ #0 " − 1 1 T v¯2,T (θF ) U 2 PF PF0 U − 2 1 √ G = 4 T v¯2,T (θ∗G ) 0n×n
#" √ F # T v¯2,T (θ ∗F ) √ G + op (1). (A.8) T v¯2,T (θ∗G )
0n×n 1
1
−U − 2 PG PG0 U − 2
From Assumptions A, B and C in the paper, we have " √ # F (θ ∗ ) T v¯2,T F A √ G ∼ N (02n , 4S) . ∗ T v¯2,T (θG )
(A.9)
Hence, 2 T (ˆδ F
2 A − ˆδ G ) ∼
"
1 2
0
zS
1
1
U − 2 PF PF0 U − 2
0n×n −U
0n×n
− 12
PG PG0 U
− 12
#
1
S 2 z,
(A.10)
where z ∼ N (02n, I2n ). Furthermore, the nonzero eigenvalues of S
1 2
"
1
1
U − 2 PF PF0 U − 2 0n×n
are the same as the eigenvalues of the matrix # " 1 PF0 U − 2 0(n−k1 )×n 1
PG0 U − 2
0(n−k2 )×n
#
0n×n
S
−U
− 12
"
U − 2 PF
PG PG0 U
1
0n×(n−k1 )
− 12
1
S2
(A.11)
0n×(n−k2 ) 1
−U − 2 PG
#
.
(A.12)
Then, it follows that 2 2 A T (ˆ δF − ˆδ G ) ∼ F2n−k1 −k2 (ξ),
(A.13)
where the ξ i ’s are the eigenvalues of the matrix "
1
1
PF0 U − 2 SF U − 2 PF 1
1
PG0 U − 2 SGF U − 2 PF
1
1
−PF0 U − 2 SFG U − 2 PG 1
1
−PG0 U − 2 SG U − 2 PG
This completes the proof of part (a). 13
#
.
(A.14)
(b) Using the result in part (b) of Lemma 1 in the paper, it can be shown that when λF = λG = 0n , √ where Σλ˜ F G =
"
A
˜ FG ∼ N (02n−k −k , Σ˜ ), Tλ 1 2 λF G 1
1
PF0 U − 2 SF U − 2 PF 1
1
PG0 U − 2 SGF U − 2 PF
(A.15)
1
1
PF0 U − 2 SFG U − 2 PG 1
1
PG0 U − 2 SG U − 2 PG
#
.
(A.16)
ˆ ˜ is a consistent estimator of Σ˜ , we have Using the fact that Σ λF G λF G A 2 ˜0 Σ ˆ −1 λ ˜ FG ∼ LMλ˜ F G = T λ χ2n−k1 −k2 . FG λ ˜ FG
This completes the proof of part (b).
14
(A.17)
References Andrews, D. W. K., 1991, Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858. Golden, R. M., 2003, Discrepancy risk model selection test theory for comparing possibly misspecified or nonnested models. Psychometrika 68, 229–249. Hansen, L. P., J. C. Heaton and E. G. J. Luttmer, 1995, Econometric evaluation of asset pricing models. Review of Financial Studies 8, 237–274. Jagannathan, R. and Z. Wang, 1996, The conditional CAPM and the cross-section of expected returns. Journal of Finance 51, 3–53. Marcellino, M. and B. Rossi, 2008, Model selection for nested and overlapping nonlinear, dynamic and possibly mis-specified models. Oxford Bulletin of Economics and Statistics 70, 867–893. Newey, W. K. and K. D. West, 1987, A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 55, 703–708. Parker, J. A. and C. Julliard, 2005, Consumption risk and the cross section of expected returns. Journal of Political Economy 113, 185–222.
15
Table 5. Model selection tests for nested models Panel A: Pairwise model comparison tests: Linear models UFT T 120 240 360 480 600
10% 0.114 0.122 0.122 0.120 0.117
Size 5% 0.056 0.064 0.066 0.065 0.061
1% 0.010 0.014 0.014 0.014 0.014
UVT 10% 0.310 0.452 0.565 0.660 0.738
Power 5% 0.188 0.314 0.423 0.521 0.612
1% 0.049 0.116 0.182 0.255 0.333
10% 0.000 0.005 0.023 0.040 0.050
Size 5% 0.000 0.001 0.006 0.013 0.019
1% 0.000 0.000 0.000 0.001 0.002
10% 0.006 0.075 0.219 0.353 0.472
Power 5% 0.001 0.020 0.089 0.179 0.272
1% 0.000 0.001 0.008 0.028 0.057
10% 0.002 0.030 0.096 0.182 0.275
Power 5% 0.000 0.008 0.033 0.077 0.136
1% 0.000 0.000 0.002 0.009 0.019
RVT T 120 240 360 480 600
10% 0.061 0.068 0.075 0.079 0.081
Size 5% 0.020 0.025 0.030 0.034 0.036
1% 0.001 0.002 0.003 0.004 0.004
10% 0.173 0.288 0.397 0.499 0.592
Power 5% 0.075 0.147 0.225 0.308 0.391
1% 0.008 0.024 0.047 0.078 0.114
Panel B: Pairwise model comparison tests: Nonlinear models UFT T 120 240 360 480 600
10% 0.082 0.118 0.121 0.117 0.111
Size 5% 0.033 0.059 0.063 0.061 0.057
1% 0.003 0.012 0.015 0.015 0.014
UVT 10% 0.254 0.426 0.527 0.612 0.686
Power 5% 0.132 0.285 0.381 0.467 0.547
1% 0.021 0.095 0.154 0.210 0.274
10% 0.000 0.001 0.002 0.006 0.010
Size 5% 0.000 0.000 0.000 0.001 0.002
1% 0.000 0.000 0.000 0.000 0.000
RVT T 120 240 360 480 600
10% 0.025 0.041 0.047 0.048 0.047
Size 5% 0.006 0.012 0.015 0.016 0.016
1% 0.000 0.001 0.001 0.001 0.001
10% 0.113 0.242 0.325 0.395 0.463
Power 5% 0.041 0.120 0.185 0.241 0.299
1% 0.004 0.022 0.043 0.066 0.093
The table presents the empirical size and power of pairwise model comparison tests for nested linear (Panel A) and nonlinear (Panel B) models. The tests considered are the unrestricted weighted chi-squared test (UFT) in (11), the unrestricted variance test (UVT) in (13), and the restricted variance test (RVT) in (14). We report results for different levels of significance (10%, 5% and 1% levels) and for different values of the number of time series observations (T ) using 100,000 simulations, assuming that the factors and the gross returns (continuously compounded gross returns in the nonlinear case) are generated from a multivariate normal distribution.
16
Table 6. Model selection tests for non-nested models Panel A: Pairwise model comparison tests: Linear models UFT T 120 240 360 480 600
10% 0.076 0.089 0.094 0.097 0.097
Size 5% 0.036 0.045 0.048 0.049 0.049
1% 0.006 0.009 0.010 0.010 0.010
UVT 10% 0.416 0.721 0.870 0.940 0.973
Power 5% 0.326 0.661 0.838 0.923 0.964
1% 0.171 0.527 0.760 0.882 0.943
10% 0.000 0.001 0.011 0.026 0.038
Size 5% 0.000 0.000 0.002 0.007 0.013
1% 0.000 0.000 0.000 0.000 0.001
10% 0.008 0.076 0.226 0.383 0.531
Power 5% 0.002 0.019 0.082 0.169 0.274
1% 0.000 0.001 0.005 0.018 0.040
10% 0.000 0.001 0.030 0.144 0.319
Power 5% 0.000 0.000 0.004 0.038 0.129
1% 0.000 0.000 0.000 0.001 0.008
RVT T 120 240 360 480 600
10% 0.038 0.049 0.058 0.066 0.070
Size 5% 0.010 0.015 0.021 0.025 0.027
1% 0.000 0.001 0.001 0.002 0.003
10% 0.116 0.209 0.324 0.451 0.577
Power 5% 0.039 0.081 0.145 0.224 0.319
1% 0.002 0.006 0.016 0.031 0.053
Panel B: Pairwise model comparison tests: Nonlinear models UFT T 120 240 360 480 600
10% 0.044 0.084 0.096 0.100 0.100
Size 5% 0.015 0.041 0.050 0.053 0.053
1% 0.001 0.008 0.012 0.013 0.013
UVT 10% 0.107 0.226 0.294 0.336 0.378
Power 5% 0.048 0.135 0.189 0.228 0.265
1% 0.005 0.036 0.061 0.082 0.104
10% 0.000 0.000 0.001 0.005 0.013
Size 5% 0.000 0.000 0.000 0.001 0.002
1% 0.000 0.000 0.000 0.000 0.000
RVT T 120 240 360 480 600
10% 0.014 0.026 0.033 0.037 0.038
Size 5% 0.003 0.007 0.009 0.011 0.012
1% 0.000 0.000 0.000 0.001 0.001
10% 0.068 0.203 0.326 0.436 0.535
Power 5% 0.021 0.091 0.176 0.266 0.358
1% 0.001 0.010 0.033 0.065 0.111
The table presents the empirical size and power of pairwise model comparison tests for overlapping linear (Panel A) and nonlinear (Panel B) models. The tests considered are the unrestricted weighted chi-squared test (UFT) in (11), the unrestricted variance test (UVT) in (13), and the restricted variance test (RVT) in (16). We report results for different levels of significance (10%, 5% and 1% levels) and for different values of the number of time series observations (T ) using 100,000 simulations, assuming that the factors and the gross returns (continuously compounded gross returns in the nonlinear case) are generated from a multivariate normal distribution.
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Table 7. Joint tests of correct specification
Panel A: Pairwise model comparison tests: Linear models RFT
LM
T
10%
Size 5%
1%
10%
Power 5%
1%
10%
Size 5%
1%
10%
Power 5%
1%
120 240 360 480 600
0.148 0.127 0.120 0.116 0.113
0.080 0.067 0.062 0.058 0.058
0.018 0.014 0.013 0.013 0.013
0.387 0.493 0.555 0.599 0.632
0.289 0.407 0.475 0.525 0.563
0.138 0.254 0.331 0.389 0.436
0.002 0.002 0.003 0.004 0.005
0.000 0.001 0.001 0.001 0.001
0.000 0.000 0.000 0.000 0.000
0.660 1.000 1.000 1.000 1.000
0.456 0.999 1.000 1.000 1.000
0.116 0.987 1.000 1.000 1.000
10% 0.505 0.997 1.000 1.000 1.000
Power 5% 0.319 0.992 1.000 1.000 1.000
1% 0.087 0.950 1.000 1.000 1.000
UFT T 120 240 360 480 600
10% 0.068 0.079 0.085 0.090 0.091
Size 5% 0.026 0.034 0.037 0.040 0.043
1% 0.002 0.004 0.005 0.007 0.007
10% 0.179 0.313 0.396 0.461 0.507
Power 5% 0.097 0.214 0.301 0.366 0.419
1% 0.019 0.076 0.146 0.206 0.260
UVT T 120 240 360 480 600
10% 0.001 0.002 0.006 0.013 0.020
Size 5% 0.000 0.000 0.001 0.003 0.005
1% 0.000 0.000 0.000 0.000 0.000
RVT 10% 0.016 0.524 0.966 0.998 1.000
Power 5% 0.004 0.245 0.898 0.991 0.999
1% 0.000 0.016 0.499 0.920 0.990
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10% 0.003 0.014 0.025 0.036 0.044
Size 5% 0.001 0.004 0.008 0.013 0.017
1% 0.000 0.000 0.001 0.001 0.002
Table 7 (continued). Joint tests of correct specification Panel B: Pairwise model comparison tests: Nonlinear models RFT T 120 240 360 480 600
10% 0.092 0.097 0.100 0.098 0.100
Size 5% 0.042 0.046 0.048 0.049 0.050
1% 0.006 0.008 0.009 0.009 0.009
LM 10% 0.309 0.458 0.535 0.587 0.625
Power 5% 0.214 0.366 0.453 0.511 0.552
1% 0.083 0.211 0.305 0.372 0.420
10% 0.002 0.002 0.002 0.004 0.005
Size 5% 0.000 0.000 0.001 0.001 0.002
1% 0.000 0.000 0.000 0.000 0.000
10% 0.694 1.000 1.000 1.000 1.000
Power 5% 0.499 0.999 1.000 1.000 1.000
1% 0.147 0.992 1.000 1.000 1.000
10% 0.413 0.995 1.000 1.000 1.000
Power 5% 0.235 0.984 1.000 1.000 1.000
1% 0.049 0.904 0.999 1.000 1.000
UFT T 120 240 360 480 600
10% 0.046 0.069 0.081 0.083 0.088
Size 5% 0.014 0.028 0.035 0.039 0.041
1% 0.000 0.002 0.004 0.006 0.006
10% 0.129 0.284 0.381 0.447 0.494
Power 5% 0.054 0.183 0.279 0.350 0.401
1% 0.004 0.053 0.124 0.184 0.238
UVT T 120 240 360 480 600
10% 0.000 0.001 0.003 0.008 0.015
Size 5% 0.000 0.000 0.000 0.001 0.004
1% 0.000 0.000 0.000 0.000 0.000
RVT 10% 0.003 0.244 0.951 0.997 1.000
Power 5% 0.001 0.049 0.814 0.988 0.999
1% 0.000 0.001 0.157 0.851 0.985
10% 0.002 0.011 0.022 0.031 0.040
Size 5% 0.000 0.003 0.006 0.010 0.014
1% 0.000 0.000 0.000 0.001 0.001
The table presents the empirical size and power of pairwise model comparison tests for overlapping linear (Panel A) and nonlinear (Panel B) correctly specified distinct SDFs. The tests considered are the restricted weighted chisquared test (RFT) in part (a) of Lemma B.2, the chi-squared test (LM) in part (b) of Lemma B.2, the unrestricted weighted chi-squared test (UFT) in (11), the unrestricted variance test (UVT) in (13), and the restricted variance test (RVT) in (16). We report results for different levels of significance (10%, 5% and 1% levels) and for different values of the number of time series observations (T ) using 100,000 simulations, assuming that the factors and the gross returns (continuously compounded gross returns in the nonlinear case) are generated from a multivariate normal distribution.
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Table 8. t-tests under model misspecification and specification tests Panel A: t-tests under potentially misspecified models Size (rmkt ) T 120 240 360 480 600
10% 0.143 0.139 0.133 0.129 0.126
5% 0.079 0.077 0.074 0.070 0.069
Size (4cndur )
1% 0.018 0.019 0.018 0.018 0.017
10% 0.093 0.092 0.094 0.095 0.097
5% 0.043 0.043 0.045 0.045 0.049
1% 0.006 0.006 0.008 0.008 0.009
Size (4cdur ) 10% 0.092 0.094 0.095 0.097 0.099
5% 0.043 0.044 0.046 0.047 0.049
1% 0.007 0.007 0.008 0.009 0.009
Panel B: t-tests under correctly specified models Size (rmkt ) T 120 240 360 480 600
10% 0.232 0.266 0.282 0.291 0.298
5% 0.150 0.181 0.198 0.208 0.216
Size (4cndur )
1% 0.050 0.074 0.088 0.098 0.103
10% 0.287 0.330 0.358 0.375 0.390
5% 0.193 0.236 0.265 0.287 0.301
1% 0.072 0.102 0.128 0.147 0.162
Size (4cdur ) 10% 0.292 0.343 0.375 0.399 0.412
5% 0.198 0.247 0.282 0.308 0.321
1% 0.074 0.110 0.140 0.162 0.181
Panel C: Specification tests HJ-distance test Size
LM test Power
Size
Power
T
10%
5%
1%
10%
5%
1%
10%
5%
1%
10%
5%
1%
120 240 360 480 600
0.280 0.165 0.139 0.126 0.119
0.180 0.093 0.074 0.067 0.061
0.063 0.024 0.017 0.014 0.014
0.991 1.000 1.000 1.000 1.000
0.983 0.999 1.000 1.000 1.000
0.948 0.997 1.000 1.000 1.000
0.136 0.114 0.110 0.107 0.104
0.065 0.056 0.054 0.053 0.051
0.010 0.011 0.010 0.010 0.010
0.989 1.000 1.000 1.000 1.000
0.974 0.999 1.000 1.000 1.000
0.897 0.998 1.000 1.000 1.000
The table presents the empirical size of t-tests of H0 : γ i = 0 in Lemma 1 in the paper and the empirical size and power of the conventional HJ-distance test in part (a) of Lemma B.1 and of the LM test in Theorem 1 in the paper. The various t-ratios are compared to the critical values from a standard normal distribution. We report results for different levels of significance (10%, 5% and 1% levels) and for different values of the number of time series observations (T ) using 100,000 simulations, assuming that the factors and the gross returns are generated from a multivariate t-distribution with eight degrees of freedom.
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Table 9. Model selection tests for nested and non-nested SDFs Panel A: Pairwise nested model comparison tests Weighted χ2 test T 120 240 360 480 600
10% 0.106 0.105 0.105 0.105 0.105
Size 5% 0.048 0.049 0.050 0.051 0.051
1% 0.007 0.008 0.008 0.009 0.009
10% 0.362 0.516 0.624 0.706 0.770
Wald test Power 5% 0.218 0.355 0.464 0.554 0.633
1% 0.053 0.116 0.185 0.256 0.327
10% 0.132 0.131 0.131 0.128 0.126
Size 5% 0.069 0.070 0.069 0.069 0.068
1% 0.015 0.015 0.015 0.016 0.016
Panel B: Multiple nested model comparison test Size Power 10% 5% 1% 10% 5% 0.110 0.053 0.010 0.398 0.269 0.112 0.055 0.011 0.584 0.454 0.116 0.060 0.012 0.704 0.589 0.115 0.059 0.012 0.789 0.691 0.114 0.058 0.013 0.846 0.766
T 120 240 360 480 600
10% 0.488 0.661 0.755 0.820 0.866
Power 5% 0.351 0.538 0.652 0.733 0.794
1% 0.146 0.297 0.420 0.519 0.601
Power 5% 0.628 0.935 0.992 0.999 1.000
1% 0.366 0.800 0.963 0.994 0.999
Power 5% 0.250 0.378 0.501 0.601 0.686
1% 0.093 0.171 0.270 0.367 0.458
1% 0.099 0.227 0.354 0.469 0.564
Panel C: Pairwise non-nested model comparison tests Weighted χ2 test T 120 240 360 480 600
10% 0.087 0.089 0.092 0.094 0.094
Size 5% 0.040 0.042 0.044 0.045 0.046
1% 0.007 0.007 0.008 0.008 0.008
10% 0.629 0.862 0.942 0.973 0.987
Wald test Power 5% 0.536 0.825 0.927 0.966 0.984
1% 0.331 0.712 0.880 0.946 0.975
10% 0.102 0.098 0.100 0.101 0.100
Size 5% 0.050 0.047 0.049 0.050 0.050
1% 0.010 0.009 0.009 0.010 0.010
10% 0.752 0.969 0.997 1.000 1.000
Panel D: Pairwise (p = 1) and multiple (p = 2) model comparison tests p=1 p=2 T 120 240 360 480 600
10% 0.135 0.111 0.105 0.103 0.103
Size 5% 0.068 0.054 0.051 0.050 0.049
1% 0.011 0.007 0.007 0.007 0.008
10% 0.434 0.597 0.711 0.792 0.851
Power 5% 0.314 0.467 0.590 0.686 0.761
1% 0.132 0.237 0.351 0.452 0.543
10% 0.134 0.107 0.101 0.100 0.100
Size 5% 0.065 0.049 0.046 0.047 0.048
1% 0.010 0.006 0.006 0.007 0.007
10% 0.364 0.507 0.625 0.718 0.789
The table presents the empirical size and power of pairwise and multiple model comparison tests for nested (Panels A and B) and non-nested (Panels C and D) models. In Panel A, we report results for the weighted chi-squared test and the Wald test in parts (a) and (b) of Theorem 2 in the paper, respectively. Panel B is for the Wald test for multiple nested model comparison analyzed in Section 3.2 of the paper. In Panel C, we report results for the weighted chi-squared test and the Wald test in parts (a) and (b) of Theorem 3 in the paper, respectively. Panel D is for the pairwise (p = 1) and multiple (p = 2) model comparison tests for non-nested misspecified (distinct) SDFs described in Sections 3.1 and 3.2 of the paper, respectively. We report results for different levels of significance (10%, 5% and 1% levels) and for different values of the number of time series observations (T ) using 100,000 simulations, assuming that the factors and the gross returns are generated from a multivariate t-distribution with eight degrees of freedom.
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